A fundamental problem in science is identifying underlying patterns of complex systems in the form of concise mathematical formulas. Current Artificial Intelligence (AI)-based methods have shown strong performance in single-scale systems, yet remain limited in identifying scale-specific formulas in multiscale complex systems. We present Deflex, an end-to-end AI method to automatically extract multiscale formulas with potentially different forms, including invariants and distributions, from complex systems. Deflex consists of two subsystems named Deflexformer and Deflexpressor. Deflexpressor is a lambda-calculus symbolic regression model for higher-order formulas. Deflexformer is a decomposable deep energy model for learning unified representations across scales. Deflexpressor generates synthetic data to pre-train Deflexformer, which then guides formula discovery by decoupling multiscale latent relationships. Across six representative complex systems with diverse behaviors, Deflex achieves up to 7-fold higher efficiency than the state-of-the-art methods while enabling automated multiscale discovery. Our work could be a useful tool for scientific discovery across disciplines.
Deriving governing equations from empirical observations is a longstanding challenge in science. Although artificial intelligence (AI) has demonstrated substantial capabilities in function approximation, the discovery of explainable and extrapolatable equations remains a fundamental limitation of modern AI, posing a central bottleneck for AI-driven scientific discovery. Here, we present machine collective intelligence, a unified paradigm that integrates two fundamental yet distinct traditions in computational intelligence--symbolism and metaheuristics--to enable autonomous and evolutionary discovery of governing equations. It orchestrates multiple reasoning agents to evolve their symbolic hypotheses through coordinated generation, evaluation, critique, and consolidation, enabling scientific discovery beyond single-agent inference. Across scientific systems governed by deterministic, stochastic, or previously uncharacterized dynamics, machine collective intelligence autonomously recovered the underlying governing equations without relying on hand-crafted domain knowledge. Furthermore, the resulting equations reduced extrapolation error by up to six orders of magnitude relative to deep neural networks, while condensing 0.5-1 million model parameters into just 5-40 interpretable parameters. This study marks an important shift in AI toward the autonomous discovery of principled scientific equations.
Scientific equation discovery must combine broad domain priors with strict numerical testing. Symbolic regression supplies numerical grounding but faces a combinatorial search space, whereas many language-model systems ask the model to propose or select formulas directly. We test a different division of labour. We compare role specifications in which the language model acts as equation author, candidate decider or search controller, alongside end-to-end language-model and purely numerical baselines. In the controller setting we propose here, implemented as LLM-PySR, language models specify variables, operators, transformations and search depth; symbolic regression enumerates and fits expressions; and deterministic metrics govern retention. Across 74 AI-Feynman equations and seven complex formula-recovery tasks, search control achieved the strongest observed balance of accuracy, complexity, stability and cost. On an independent battery dataset, LLM-PySR identified a compact piecewise-linear relation between early voltage-curve displacement and cycle life. The results suggest that language models should shape hypothesis exploration rather than decide which equations survive.
Scientific equation discovery has long been central to scientific progress, proceeding through iterative cycles of hypothesis generation, observational testing, and refinement under scientific constraints. As LLM capabilities advance and their role in AI for Science expands, it remains an open problem whether they can genuinely discover scientific laws and how this ability should be evaluated. Existing evaluations, however, often either simplify discovery through synthetic settings or reuse published targets that may already be familiar to LLMs. We therefore introduce SCILAWS-BENCH, a benchmark for scientific law discovery built from published research and real scientific data. It comprises 118 problems drawn from 381 scientific papers, covering 291 candidate laws and roughly 8M real data points across six scientific disciplines. Each problem is instantiated in two complementary settings: (1) SCILAWS-REAL asks models to propose laws from fixed real observations and evaluates held-out predictive fit and scientific validity derived from the source literature, and (2) SCILAWS-PARALLEL asks models to actively query residual-calibrated worlds and recover synthesized hidden laws derived from published forms. This two-setting task design preserves each problem's scientific context while separately evaluating fixed-record law discovery and active recovery of a newly synthesized hidden law. We find that predictive fit can diverge from scientific validity, memorization shapes whether models reproduce or move beyond published formulas, and our best-of-N study reveals a selection bottleneck. Our work provides a paper-grounded benchmark and new empirical perspectives for evaluating AI for scientific discovery. Project page: https://yiyihum.github.io/SciLaws-Bench